Prediction Error Covariance Research Articles

Machine Learning-Enhanced Probabilistic Validation Techniques with Minimal Data Exposure in Distributed Systems

This article studies the distributed state estimation issue for complex networks with nonlinear uncertainty. The extended state approach is used to deal with the nonlinear uncertainty. The distributed state predictor is designed based on the extended state system model, and the distributed state estimator is designed by using the measurement of the corresponding node. The prediction error and the estimation error are derived. The prediction error covariance (PEC) is obtained in terms of the recursive Riccati equation, and the upper bound of the PEC is minimized by designing an optimal estimator gain. With the vectorization approach, a sufficient condition concerning stability of the upper bound is developed. Finally, a numerical example is presented to illustrate the effectiveness of the designed extended state estimator.

Encryption–decryption-based state estimation for nonlinear complex networks subject to coupled perturbation

This paper discusses the encryption-decryption-based state estimation (EDBSE) issue for coupled perturbation complex networks (CPCNs) in the framework of the Kalman-type filtering scheme. A uniform distributed random variable is employed to characterize the coupled perturbation among different network units. A uniform-quantization-dependent encryption-decryption (UQDED) scheme is considered here to orchestrate the transmitted data. A novel EDBSE approach is developed such that the upper bounds of prediction error (PE) covariance (PEC) and estimation error (EE) covariance (EEC) can be derived by resolving Riccati-like difference equations and the estimation parameter (EP) is determined by minimizing the trace of the upper bound of EEC. Furthermore, a uniformly bounded condition is elaborated to evaluate the algorithm performance of EDBSE. Finally, an illustrative example is conducted to verify the validity of the introduced EDBSE method.

State estimation for linear discrete-time systems with binary-valued quantized innovations against data tampering attacks

In today’s information era, cyber–physical systems (CPS) have been widely used in many industrial and infrastructure fields, and the study of security problems of CPS has become a crucial issue. For linear discrete-time systems, this paper investigates the state estimation problem based on the quantized innovations and transmission data tampering scenarios. From the attacker’s view, by designing the tampering matrix and combining the property of the error function, the optimal attack strategy in the sense of maximizing the prediction error covariance matrix is given. From the defender’s view, the optimal defense scheme based on multiplicative compensation is given in the case of a known attack strategy. Moreover, in the case of unknown attack strategies, a joint estimation algorithm is designed to give the optimal defense scheme subject to error constraints. In this paper, state estimators in recursive form are given for all above cases, and the algorithms are proved to be consistent. The theoretical results obtained are verified by simulations.

Confidence intervals for validation statistics with data truncation in genomic prediction

BackgroundValidation by data truncation is a common practice in genetic evaluations because of the interest in predicting the genetic merit of a set of young selection candidates. Two of the most used validation methods in genetic evaluations use a single data partition: predictivity or predictive ability (correlation between pre-adjusted phenotypes and estimated breeding values (EBV) divided by the square root of the heritability) and the linear regression (LR) method (comparison of “early” and “late” EBV). Both methods compare predictions with the whole dataset and a partial dataset that is obtained by removing the information related to a set of validation individuals. EBV obtained with the partial dataset are compared against adjusted phenotypes for the predictivity or EBV obtained with the whole dataset in the LR method. Confidence intervals for predictivity and the LR method can be obtained by replicating the validation for different samples (or folds), or bootstrapping. Analytical confidence intervals would be beneficial to avoid running several validations and to test the quality of the bootstrap intervals. However, analytical confidence intervals are unavailable for predictivity and the LR method.ResultsWe derived standard errors and Wald confidence intervals for the predictivity and statistics included in the LR method (bias, dispersion, ratio of accuracies, and reliability). The confidence intervals for the bias, dispersion, and reliability depend on the relationships and prediction error variances and covariances across the individuals in the validation set. We developed approximations for large datasets that only need the reliabilities of the individuals in the validation set. The confidence intervals for the ratio of accuracies and predictivity were obtained through the Fisher transformation. We show the adequacy of both the analytical and approximated analytical confidence intervals and compare them versus bootstrap confidence intervals using two simulated examples. The analytical confidence intervals were closer to the simulated ones for both examples. Bootstrap confidence intervals tend to be narrower than the simulated ones. The approximated analytical confidence intervals were similar to those obtained by bootstrapping.ConclusionsEstimating the sampling variation of predictivity and the statistics in the LR method without replication or bootstrap is possible for any dataset with the formulas presented in this study.

Assessment, Specification, and Validation of a Geolocation System's Accuracy and Predicted Accuracy

This article presents recommendations and corresponding detailed procedures for the assessment of a geolocation system's accuracy, as well as the specification of accuracy requirements and their subsequent validation when they are available. Applicable metrics and related processing are based on samples of corresponding geolocation errors. This article also presents similar recommendations for the predicted accuracy of a geolocation system, based on samples of geolocation error, as well as corresponding predicted error covariance matrices associated with the geolocations. Reliable error covariance matrices enable optimal use of a geolocation system's products, such as the optimal fusion of multiple geolocations or multiple products for higher confidence and increased accuracy. The recommendations presented in this article enable reliable estimates of accuracy and reliable predicted accuracies, both of which are critical to many geolocation-based applications. The recommendations associated with predicted accuracy are also relatively new and innovative.

Collaborative Parameter Estimation of Multiple Unmanned Surface Vessels: A Robust Distributed Estimator-Based Approach

In this paper, the collaborative parameter estimation of multiple unmanned surface vessels (USVs) with model structure uncertainties is studied. The considered parameter estimation problem is first converted into a distributed state and parameter joint estimation problem. Then, the robust distributed estimator is constructed to handle the inevitable model structure uncertainties, and the upper bounds of prediction and estimation error covariance matrices are derived respectively. Additionally, the upper bound of the estimation error covariance matrix is minimized by designing appropriate estimator gains. Finally, the advantages of the proposed distributed parameter estimation approach from the perspectives of information interaction and consideration of model structure uncertainties are verified via simulations and practical experiments.

Remote State Estimation With Posterior-Based Stochastic Event-Triggered Schedule

This paper aims to study the state estimation problem under the stochastic event-triggered (SET) schedule. A posterior-based SET mechanism is proposed, which determines whether to transmit data by the effect of the measurement on the posterior estimate. Since this SET mechanism considers the whole posterior probability density function, it has better information screening capability and utilization than the existing SET mechanisms that only consider the first-order moment information of measurement and prior estimate. Then, based on the proposed SET mechanism, the corresponding exact minimum mean square error estimator is derived by Bayes rule. Moreover, the prediction error covariance of the estimator is proved to be bounded under moderate conditions. Meanwhile, the upper and lower bounds on the average communication rate are also analyzed. Finally, two different systems are employed to show the effectiveness and advantages of the proposed methods.

How Not to Make the Joint Extended Kalman Filter Fail with Unstructured Mechanistic Models.

The unstructured mechanistic model (UMM) allows for modeling the macro-scale of a phenomenon without known mechanisms. This is extremely useful in biomanufacturing because using the UMM for the joint estimation of states and parameters with an extended Kalman filter (JEKF) can enable the real-time monitoring of bioprocesses with unknown mechanisms. However, the UMM commonly used in biomanufacturing contains ordinary differential equations (ODEs) with unshared parameters, weak variables, and weak terms. When such a UMM is coupled with an initial state error covariance matrix P(t=0) and a process error covariance matrix Q with uncorrelated elements, along with just one measured state variable, the joint extended Kalman filter (JEKF) fails to estimate the unshared parameters and state simultaneously. This is because the Kalman gain corresponding to the unshared parameter remains constant and equal to zero. In this work, we formally describe this failure case, present the proof of JEKF failure, and propose an approach called SANTO to side-step this failure case. The SANTO approach consists of adding a quantity to the state error covariance between the measured state variable and unshared parameter in the initial P(t = 0) of the matrix Ricatti differential equation to compute the predicted error covariance matrix of the state and prevent the Kalman gain from being zero. Our empirical evaluations using synthetic and real datasets reveal significant improvements: SANTO achieved a reduction in root-mean-square percentage error (RMSPE) of up to approximately 17% compared to the classical JEKF, indicating a substantial enhancement in estimation accuracy.

Modeling and Forecasting Closing Prices of some Coal Mining Companies in Indonesia by Using the VAR(3)-BEKK GARCH(1,1) Model

Today, coal is the main source of energy in both developed and developing countries. The use of coal fuel for power generation and industry continues to increase. This research will discuss the closing price relationship model for the share prices of two coal companies in Indonesia, namely ABM and IND_E, from January 2018 to July 2023. The modeling used is a multivariate time series approach. From the results of the data analysis, the best model that fits the data is the VAR(3)-BEKK GARCH(1,1). Based on this best model, further analysis of Granger causality, impulse response function (IRF), and forecasting for the next 30 periods as well as the proportion of prediction error covariance are discussed.

A Variational Bayesian Truncated Adaptive Filter for Uncertain Systems with Inequality Constraints

In this paper, a variational Bayesian (VB) truncated adaptive filter for uncertain systems with inequality constraints is proposed. By choosing the skew‐t and inverse Wishart distributions as the prior information of the measurement noise and predicted error covariance matrix, the state vector, the predicted error covariance matrix, and noise parameters are inferred and approximated by using the VB method. To achieve the inequality‐constrained estimation, the constrained state is computed by truncating the probability density function (PDF) of the estimated state after the variational update stage; the mean and covariance of the constrained state are the first and second moments of the truncated PDF. Considering the model uncertainties where the system dynamics are unpredictable, a multiple model VB truncated adaptive filter is proposed in the interacting multiple model framework. The performances of the proposed algorithms are evaluated via the target tracking simulations and the robot positioning experiments. Results show that the proposed algorithms improve estimation accuracy compared with the existing adaptive filters when the states suffer inequality constraints.

Estimation-free Prediction Algorithms

For Time-varying, Time-invariant, and steady-state systems, Kalman Filter can be implemented as a prediction algorithm, since it produces the state prediction and the corresponding prediction error covariance matrix via the state estimation and the corresponding estimation error covariance matrix. Lainiotis Filter is equivalent to Kalman Filter and can be used to compute the prediction. In this paper, for Time-varying, Time-invariant and steady state systems, estimation-free Prediction Algorithms are derived via Kalman and Lainiotis filters; they are equivalent and compute iteratively the prediction and the corresponding prediction error covariance matrix. The estimation and the corresponding estimation error covariance matrix are not needed and are not computed. The proposed estimation-free prediction algorithms are faster than the Kalman filter.

Chandrasekhar-type Algorithms with Gain Elimination

Chandrasekhar-type algorithms are associated with the Riccati equation emanating from the Kalman filter in linear systems which describe the relationship between the n-dimensional state and the m-dimensional measurement. The traditional Chandrasekhar-type algorithms use the Kalman filter gain to compute the prediction error covariance. In this paper, two variations of Chandrasekhar-type algorithms eliminating the Kalman filter gain are proposed. The proposed Chandrasekhar-type algorithms with gain elimination may be faster than the traditional Chandrasekhar-type algorithms, depending on the model dimensions.

Attitude Determination System for a Cubesat Experiencing Eclipse.

In the context of Kalman filters, the predicted error covariance matrix Pk+1 and measurement noise covariance matrix R are used to represent the uncertainty of state variables and measurement noise, respectively. However, in real-world situations, these matrices may vary with time due to measurement faults. To address this issue in CubeSat attitude estimation, an adaptive extended Kalman filter has been proposed that can dynamically estimate the predicted error covariance matrix and measurement noise covariance matrix using an expectation-maximization approach. Simulation experiments have shown that this algorithm outperforms existing methods in terms of attitude estimation accuracy, particularly in sunlit and shadowed phases of the orbit, with the same filtering parameters and initial conditions.

Distributed Extended State Estimation for Complex Networks With Nonlinear Uncertainty.

This article studies the distributed state estimation issue for complex networks with nonlinear uncertainty. The extended state approach is used to deal with the nonlinear uncertainty. The distributed state predictor is designed based on the extended state system model, and the distributed state estimator is designed by using the measurement of the corresponding node. The prediction error and the estimation error are derived. The prediction error covariance (PEC) is obtained in terms of the recursive Riccati equation, and the upper bound of the PEC is minimized by designing an optimal estimator gain. With the vectorization approach, a sufficient condition concerning stability of the upper bound is developed. Finally, a numerical example is presented to illustrate the effectiveness of the designed extended state estimator.

Robust filtering for spacecraft attitude estimation systems with multiplicative noises, unknown measurement disturbances and correlated noises

Unknown measurement disturbances and correlated noise caused by jitter or vibration phenomena during spacecraft operation and the complex external environment are important topics for spacecraft attitude estimation. This paper studies the problem of spacecraft attitude estimation for nonlinear systems with multiplicative noises, unknown measurement disturbances and correlated noises. In addition, a two-step prediction framework is used to complete the noise decoupling. This paper aims to design a robust filter that minimizes the upper bound of the prediction error covariance and estimation error covariance in the presence of multiplicative noises, correlated noises and unknown measurement disturbances. The prediction gain and filtering gain of the robust filter are designed to minimize the upper bound. Finally, the simulation results verify the effectiveness of the proposed filter.

Distributed Estimation and Smoothing for Linear Dynamic Systems Over Sensor Networks

This paper focuses on distributed estimation and smoothing for linear dynamic systems monitored by sensors that are organized as a network. To achieve low power consumption of sensors, the system is decomposed into several subsystems based on the observable canonical decomposition method. Each sensor in the network performs state estimation for one subsystem at most, while estimations for the majority of subsystems are obtained from the neighbors of each sensor. The optimal gains of the estimator and the smoother are pre-calculated offline, and a numerical solution is obtained by taking advantage of optimization algorithms. Furthermore, the boundedness of the prediction error covariance is guaranteed resulting from the bounded solution of a Riccati equation, and the performance of the smoother is analyzed. A numerical example is presented to validate the feasibility of the algorithm even if the sensor network suffers from network interruptions. The proposed algorithm is more robust against partial loss of predictions and recovers more rapidly, compared to some consensus-based distributed estimation algorithms.

Maximum correntropy criterion variational Bayesian adaptive Kalman filter based on strong tracking with unknown noise covariances

The performance of the current state estimation will degrade in the existence of slow-varying noise statistics. To solve the aforementioned issues, an improved strong tracking maximum correntropy criterion variational-Bayesian adaptive Kalman filter is presented in this paper. First of all, the inverse-Wishart distribution, as the conjugate-prior, is adopted to model the unknown and time-varying measurement and process noise covariances, then the noise covariances and system state are estimated via the variational Bayesian method. Secondly, the multiple fading-factors are obtained and evaluated to modify the prediction error covariance matrix to address the problems associated with inaccurate error estimation. Finally, the maximum correntropy criterion is employed to correct the filtering gain, which improves the filtering performance of the proposed algorithm. Simulation results show that the proposed filter exhibits better accuracy and convergence performance compared to other existing algorithms.

A novel variational Bayesian adaptive Kalman filter with mismatched process noise covariance matrix

AbstractThis paper proposes a novel variational Bayesian (VB) adaptive Kalman filter with mismatched process noise covariance matrix (PNCM). Firstly, this paper explains the reason why the predicted error covariance matrix (PECM) is chosen for variational inference. Secondly, compared with the earlier VB adaptive Kalman filter (VB‐AKF‐Q), the proposed filter calculate the dynamic model of the PECM with its historical estimation information. Therefore, the proposed filter can overcome the influence of mismatched PNCM on the initial value setting of PECM in the VB‐AKF‐Q. Finally, we use the evidence lower bound for the proposed filter and give the convergence criterion on this basis. Some examples with a target tracking simulation are carried out to demonstrate the superiority of the proposed filter.

Robust interacting multiple model cubature Kalman filter for nonlinear filtering with unknown non-Gaussian noise

Estimating noise covariance matrices and suppressing outliers simultaneously is a huge challenge because they are coupled. In order to solve the nonlinear filtering problem with unknown non-Gaussian noise, the maximum correntropy criterion (MCC) is used to suppress the outliers, while the variational Bayesian (VB) technique is used to estimate the one-step prediction error covariance matrix (PECM) and the measurement noise covariance matrix (MNCM), and the nonlinear problem is solved by the third-order spherical radial cubature rule. In this paper, nominal measurement is constructed to eliminate the influence of outliers on the recursive estimation of MNCM. Through these operations, the variational Bayesian and maximum correntropy based cubature Kalman filter (VBMCCKF) and the maximum correntropy cubature Kalman filter (MCCKF) are derived. The interacting multiple model (IMM) fusion framework is used to promote the collaborative work of VBMCCKF and MCCKF, thus a novel filter is proposed in this paper. The simulation verifies the validity and universal applicability of the proposed filter. The estimation error of the proposed filter in spacecraft autonomous celestial navigation is about 30% less than that of the existing filter with the best performance.

Reliabilities of estimated breeding values in models with metafounders

BackgroundReliabilities of best linear unbiased predictions (BLUP) of breeding values are defined as the squared correlation between true and estimated breeding values and are helpful in assessing risk and genetic gain. Reliabilities can be computed from the prediction error variances for models with a single base population but are undefined for models that include several base populations and when unknown parent groups are modeled as fixed effects. In such a case, the use of metafounders in principle enables reliabilities to be derived.MethodsWe propose to compute the reliability of the contrast of an individual’s estimated breeding value with that of a metafounder based on the prediction error variances of the individual and the metafounder, their prediction error covariance, and their genetic relationship. Computation of the required terms demands only little extra work once the sparse inverse of the mixed model equations is obtained, or they can be approximated. This also allows the reliabilities of the metafounders to be obtained. We studied the reliabilities for both BLUP and single-step genomic BLUP (ssGBLUP), using several definitions of reliability in a large dataset with 1,961,687 dairy sheep and rams, most of which had phenotypes and among which 27,000 rams were genotyped with a 50K single nucleotide polymorphism (SNP) chip. There were 23 metafounders with progeny sizes between 100,000 and 2000 individuals.ResultsIn models with metafounders, directly using the prediction error variance instead of the contrast with a metafounder leads to artificially low reliabilities because they refer to a population with maximum heterozygosity. When only one metafounder is fitted in the model, the reliability of the contrast is shown to be equivalent to the reliability of the individual in a model without metafounders. When there are several metafounders in the model, using a contrast with the oldest metafounder yields reliabilities that are on a meaningful scale and very close to reliabilities obtained from models without metafounders. The reliabilities using contrasts with ssGBLUP also resulted in meaningful values.ConclusionsThis work provides a general method to obtain reliabilities for both BLUP and ssGBLUP when several base populations are included through metafounders.